Why the two-pulse photon echo is not a good quantum memory protocol

We consider in this paper a two-pulse photon echo sequence in the prospect of quantum light storage. We analyze the conditions where quantum storage could be realistically performed. We simply and analytically calculate the efficiency in that limit, and clarify the role of the exactly $\pi$-rephasing pulse in the sequence. Our physical interpretation of the process is well supported by its experimental implementation in a ${\text{Tm}}^{3+}$:yttrium aluminum garnet crystal thanks to an accurate control of the rephasing pulse area. We finally address independently the fundamental limitations of the quantum fidelity. Our work allows us to point out on one side the real drawbacks of this scheme for quantum storage and on the other side its specificities which can be a source of inspiration to conceive more promising procedures with rare-earth ion doped crystals.

Figure 1

Outline of the time sequence. The signal is first absorbed in the medium. After a time $t_{12}$ a rephasing pulse of large area $A$ induces a buildup of the coherence at the time $2t_{12}$ and gives rise to the photon echo. We address the problem in the weak signal and the echo regime, which shall be satisfied in quantum field conditions.

Figure 2

Efficiency of the 2PE as a function of the rephasing pulse area $A\left(0\right)$ for different optical thickness $\alpha L$. Inset: for larger $\alpha L$, the efficiency is much larger than unity at the maximum rephasing.

Figure 3

Penetration depth $L_P$ of a strong pulse in unit of ${\alpha }^{-1}$ as a function of the incoming area. Inset: penetration of pulses with area of $\pi$ plus or minus 1%. For small-area pulses, it is simply given by an absorption law $L_P={\alpha }^{-1}$.

Figure 4

(Color online) Time sequence for $1.1\pi$-rephasing pulse. With the signal and the echo being very weak, we magnify their scale by a factor of 150 (solid red line). The intensity is given as the square of the Rabi frequency. The signal and the rephasing pulse have a duration of 2.1 and $0.8\mu \text{s}$, respectively (rms width of the Gaussian). We also represent the rephasing pulse after transmission (dashed line in Arb. units).

Figure 5

Efficiency of the echo retrieval as a function of the rephasing pulse area $A\left(0\right)$. The maximum (circle) is obtained for $A\left(0\right)=1.1\pi$, where the efficiency is 92%. This corresponds to the situation depicted in Fig. 4.